# GENERALIZED FIBONACCI NUMBERS OF THE FORM 11x2 + 1

• Ogut, Ummugulsum ;
• Keskin, Refik (Department of Mathematics, Sakarya University)
• Accepted : 2018.02.13
• Published : 2018.03.25
• 143 6

#### Abstract

Let $P{\geq}3$ be an integer and let ($U_n$) denote generalized Fibonacci sequence defined by $U_0=0$, $U_1=1$ and $U_{n+1}=PU_n-U_{n-1}$ for $n{\geq}1$. In this study, when P is odd, we solve the equation $U_n=11x^2+1$. We show that only $U_1$ and $U_2$ may be of the form $11x^2+1$.

#### Keywords

Generalized Fibonacci numbers;Generalized Lucas numbers;Congruences;Diophantine equation

#### References

1. M. A. Alekseyev and S. Tengely, On integral points on biquadratic curves and near-multiples of squares in Lucas sequences, J. Integer Seq. 17 , no. 6, Article ID 14.6.6, (2014).
2. M. A. Bennett, S. Dahmen, M. Mignotte, S. Siksek, Shifted powers in binary recurrence sequences, Math. Proc. Cambridge Philos. Soc. 158(2), (2015), 305-329. https://doi.org/10.1017/S0305004114000681
3. Y. Bugeaud, F. Luca, M. Mignotte, S. Siksek, Fibonacci numbers at most one away from a perfect power, Elem. Math. 63(2), (2008), 65-75.
4. Y. Bugeaud, F. Luca, M. Mignotte, S. Siksek, Almost powers in the Lucas sequence, J. Theor. Nombres Bordeaux, 20(3), (2008), 555-600. https://doi.org/10.5802/jtnb.642
5. Y. Bugeaud, M. Mignotte, S. Siksek, Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers, Ann. of Math. 163(3), (2006), 969-1018. https://doi.org/10.4007/annals.2006.163.969
6. J. H. E. Cohn, Squares in some recurrent sequences, Pacific Journal of Mathematics, 41 (1972), 631-646. https://doi.org/10.2140/pjm.1972.41.631
7. O. Karaatli and R. Keskin, Generalized Lucas Numbers of the form $5kx^2$ and $7kx^2$, Bulletin of the Korean Mathematical Society, 52 (2014) 1467-1480.
8. R. Keskin, Generalized Fibonacci and Lucas Numbers of the form $wx^2$ and $wx^2{\mp}1$, Bulletin of the Korean Mathematical Society, 51 (2014) 1041-1054. https://doi.org/10.4134/BKMS.2014.51.4.1041
9. M. G. Duman and U. Ogut and R. Keskin, Generalized Lucas Numbers of the form $wx^2$ and $wkx^2$, Hokkaido Mathematical Journal, (accepted).
10. R, Keskin and U. Ogut, Generalized Fibonacci Numbers of the form $wx^2+1$, Period. Math. Hung. (73) (2016), 165-178. https://doi.org/10.1007/s10998-016-0133-4
11. W. L. McDaniel, The g.c.d. in Lucas sequences and Lehmer number sequences, The Fibonacci Quarterly, 29 (1991), 24-30.
12. P. Ribenboim and W. L. McDaniel, The square terms in Lucas sequences, Journal of Number Theory, 58 (1996), 104-123. https://doi.org/10.1006/jnth.1996.0068
13. P. Ribenboim and W. L. McDaniel, Squares in Lucas sequences having an even first parameter, Colloquium Mathematicum, 78 (1998), 29-34. https://doi.org/10.4064/cm-78-1-29-34
14. P. Ribenboim, My Numbers, My Friends, Springer-Verlag New York, Inc., 2000.
15. P. Ribenboim and W. L. McDaniel, On Lucas sequence terms of the form $kx^2$, Number Theory: proceedings of the Turku symposium on Number Theory in memory of Kustaa Inkeri (Turku, 1999), de Gruyter, Berlin, 2001, 293-303.
16. Z. Siar and R. Keskin, Some new identities concerning generalized Fibonacci and Lucas numbers, Hacettepe Journal of Mathematics and Statistics, 42(3) (2013), 211-222.
17. Z. Siar and R. Keskin, The square terms in Generalized Fibonacci Sequence, Mathematika, 60 (2014), 85-100. https://doi.org/10.1112/S0025579313000193